The exact-dimensionality of \(\Phi _{\mathbf {I}}(\widetilde{\mu }^+)\)  when \(\mu ^+\)  is a Bernoulli measure on \(\Sigma _a^+\)  is proved in [1]}. Here we shall prove a more general result for \(\Phi _{\mathbf {I}}(\widetilde{\mu }^+)\)  by using the exact-dimensionality of \(\Phi _{\mathbf {I}}(\widetilde{\mu }^+_{\mathbf {i}^-})\)  for \(\mu ^{-}\) -a.e. \(\mathbf {i}^-\) , where \(\mu \)  is the natural extension of \(\mu ^+\)  to \(\Sigma _a\) , \(\mu ^-=\mu \circ (\pi ^-)^{-1}\)  and \(\mu ^+_{\mathbf {i}^-}\)  is the fibre measures on \(\Sigma _a^+\) .
